Indefinite Integrals

The Antiderivative

Definition

A function \(F\) is an antiderivative of \(f\) on an interval \(I\) if

\[F'(x) = f(x)\]

for all \(x\) in \(I\).

In other words, the phrase “\(F\) is an antiderivative of \(f\)” means the same thing as the phrase “\(f\) is the derivative of \(F\).”

Example 1

Verify a function is an antiderivative

Show that \(F(x) = 3x^5 + 4x^3 + 7x\) is an antiderivative of \(f(x) = 15x^4+12x^2+7\).

Step 1:   Verify that \(F'(x) = f(x)\).

\[\begin{align*} F'(x) &= \frac{d}{dx}\left( 3x^5 + 4x^3 + 7x \right) \\ &= 3 \cdot 5 x^4+ 4\cdot 3x^2 + 7\\ & = 15x^4+12x^2+7 \\ & = f(x) \end{align*}\]

Because \(F'(x) = f(x)\), \(F\) is an antiderivative of \(f\).

Example 2

Verify a function is an antiderivative

Show that \(G(x) = 3x^5 + 4x^3 + 7x + 71\) is an antiderivative of \(f(x) = 15x^4+12x^2+7\).

Step 1:   Verify that \(G'(x) = f(x)\).

\[\begin{align*} G'(x) &= \frac{d}{dx}\left( 3x^5 + 4x^3 + 7x + 71\right) \\ &= 3 \cdot 5 x^4+ 4\cdot 3x^2 + 7 + 0\\ & = 15x^4+12x^2+7 \\ & = f(x) \end{align*}\]

Because \(G'(x) = f(x)\), \(G\) is an antiderivative of \(f\).

Observation

In the previous two examples, the functions \(F(x)\) and \(G(x)\) were both shown to be antiderivatives of \(f(x) = 15x^4+12x^2+7\). This implies that antiderivatives are not unique. Also notice that \(G(x) = F(x) + 71\) and adding a constant to a function doesn’t change its derivative since the derivative of a constant is zero. So once we knew that \(F(x)\) was an antiderivative of \(f(x)\), we didn’t need to verify that \(G(x)\) was also an antiderivative of \(f(x)\) since it only differs from \(F(x)\) by a constant.

In general, if \(F\) is an antiderivative of \(f\), then \(F(x) + C\), where \(C\) is any constant, is also an antiderivative of \(f\). Furthermore, every antiderivative of \(f\) must be of the form \(F(x) + C\) for some constant \(C\).

The Indefinite Integral

Definition and Notation

The indefinite integral of \(f(x)\) with respect to \(x\), denoted

\[\int f(x) ~dx\]

represents the most general antiderivative of \(f\).

The symbol \(\displaystyle \int\) is called an integral sign and the symbol \(dx\) is called a differential, which indicates the variable of integration.

The Most General Antiderivative

If \(F\) is an antiderivative of \(f\), then the most general antiderivative of \(f\) is given by

\[\int f(x) ~dx = F(x) + C\]

where \(C\) is an arbitrary constant referred to as the constant of integration.

Example 3

Indefinite integral

Compute \(\displaystyle \int 15x^4+12x^2+7 ~dx.\)

Answer

From Example 1 above, we know that \(3x^5 + 4x^3 + 7x\) is an antiderivative of \(15x^4+12x^2+7\). Therefore,

\[ \int 15x^4+12x^2+7 ~dx = 3x^5 + 4x^3 + 7x + C.\]

Therefore, every antiderivative of \(15x^4+12x^2+7\) can be written as \(3x^5 + 4x^3 + 7x + C\) for some constant \(C\).